What is the Exponential Distribution?
Imagine you’re waiting for a bus. The time it takes for the next bus to arrive can vary, but the longer you wait, the more likely it is to arrive soon. The Exponential Distribution helps us understand the probability of waiting a certain amount of time before an event occurs, given that it occurs randomly and independently at a constant rate.
Exponential PDF
The probability density function (PDF) of the exponential distribution is given by:
Where:
▪ λ (lambda) is the rate parameter, representing the average number of events occurring per unit time.
â–ª x is the time elapsed since the last event.
Examples of the Exponential Distribution
Arrival Times:
â–ª Suppose buses arrive at a bus stop with an average rate of 10 buses per hour. The exponential distribution can help us calculate the probability of waiting less than 5 minutes for the next bus.
Radioactive Decay:
â–ª Radioactive atoms decay randomly over time, with the rate of decay determined by their half-life. The exponential distribution can model the time it takes for a certain percentage of atoms to decay.
Visualization
Scenario
A customer service center receives calls at an average rate of 5 calls per hour. The time between consecutive calls follows an Exponential distribution. We want to find the probability that the time until the next call is more than 15 minutes.
Solution
The Exponential distribution is often used to model the time between events in a Poisson process. It is characterized by the parameter λ, which is the rate of occurrences. In this case, the rate λ is 5 calls per hour.
The cumulative distribution function (CDF) is F(t∣λ)=1−e^(−λt)
where:
- t is the time between events.
- λ is the rate parameter.
To find the probability that the time until the next call is more than 15 minutes, we use the CDF to find the probability that it is less than or equal to 15 minutes and then subtract from 1.
Given
- λ = 5 calls per hour.
- We need to convert 15 minutes to hours because λ is given in calls per hour. Thus, 15 minutes = 15/60 = 0.25 hours.
We want P(T>0.25).
Verification Using Python
So, the probability that the time until the next call is more than 15 minutes is approximately 0.2865 or 28.65%.
Why Does This Matter?
The Exponential Distribution has applications in various fields, including queuing theory, reliability engineering, and finance. It provides insights into the timing of events and helps us make informed decisions based on probabilities.
If you enjoyed this article, feel free to follow me for more insights and updates.
LinkedIn GitHub